In a much broader paper in “Optimization Methods & Software 27,6 (2012) pp1073-1099” Bayen & Henrion consider planar compact, convex sets with support functions which are finite Fourier series.

They provide convincing numerical evidence about such sets of minimal area, a convergence theorem to the Reuleaux triangle and a square wave based suggestion for the support function of such extremal sets.

My question is much simpler, mindful of the numerous, early 20th century orbiform references, which sadly I have not read, in Bonnesen and Fenchel,

Is there a known (elementary) proof that such extremal sets with support functions which are finite Fourier series are symmetric i.e. have just a cosine series support function?